Question
Current Skill A line has this equation: \( y=\frac{1}{6} x+3 \) Write an equation for the perpendicular line that goes through \( (-5,8) \).
Ask by Dawson Gibson. in the United States
Feb 02,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The equation of the perpendicular line is \( y = -6x - 22 \).
Solution
To find the equation of the line perpendicular to \( y = \frac{1}{6}x + 3 \) that passes through the point \( (-5, 8) \), follow these steps:
1. **Determine the slope of the given line:**
The equation is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope.
\[
m_1 = \frac{1}{6}
\]
2. **Find the slope of the perpendicular line:**
The slope of a line perpendicular to another is the negative reciprocal of the original slope.
\[
m_2 = -\frac{1}{m_1} = -6
\]
3. **Use the point-slope form to find the equation:**
The point-slope form is:
\[
y - y_1 = m(x - x_1)
\]
Substituting \( m = -6 \) and the point \( (-5, 8) \):
\[
y - 8 = -6(x - (-5))
\]
\[
y - 8 = -6(x + 5)
\]
4. **Simplify to slope-intercept form:**
\[
y - 8 = -6x - 30
\]
\[
y = -6x - 22
\]
**Final Answer:**
\[
y = -6x - 22
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To find the equation of the line perpendicular to \( y=\frac{1}{6}x+3 \), we first need to determine the slope of the original line, which is \( \frac{1}{6} \). The slope of a line that is perpendicular to it is the negative reciprocal, so we calculate it as \( -6 \). Now using the point-slope form \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point \( (-5, 8) \), we plug in our values: \[ y - 8 = -6(x + 5) \] Distributing the slope: \[ y - 8 = -6x - 30 \] Now, solving for \( y \): \[ y = -6x - 30 + 8 \] \[ y = -6x - 22 \] Thus, the equation of the perpendicular line is \( y = -6x - 22 \).
